Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems

TL;DR

This paper introduces a new comprehensive theorem that characterizes intermediate operator algebras via intermediate extensions of dynamical systems, expanding the understanding of algebra inclusions.

Contribution

It establishes a third complete description theorem linking intermediate operator algebras with dynamical system extensions, complementing existing theorems.

Findings

New bijective correspondence between intermediate algebras and dynamical system extensions

Application to maximal amenability problem

Realization of intermediate operator algebra lattice

Abstract

Practically and intrinsically, inclusions of operator algebras are of fundamental interest. The subject of this paper is intermediate operator algebras of inclusions. There are two previously known theorems which naturally and completely describe all intermediate operator algebras: the Galois Correspondence Theorem and the Tensor Splitting Theorem. Here we establish the third, new complete description theorem which gives a canonical bijective correspondence between intermediate operator algebras and intermediate extensions of dynamical systems. One can also regard this theorem as a crossed product splitting theorem, analogous to the Tensor Splitting Theorem. We then give concrete applications, particularly to maximal amenability problem and a new realization result of intermediate operator algebra lattice.